A food store makes a 11–pound mixture of peanuts, almonds, and raisins. The cost of peanuts is $1.50 per pound, almonds cost $3.00 per pound, and raisins cost $1.50 per pound. The mixture calls for twice as many peanuts as almonds. The total cost of the mixture is $21.00. How much of each ingredient did the store use?
3 lbs peanuts, 6 lbs almonds, 2 lbs raisins
8 lbs peanuts, 1 lb almonds, 2 lbs raisins
6 lbs peanuts, 3 lbs almonds, 2 lbs raisins
8 lbs peanuts, 2 lbs almonds, 1 lbs raisins

- anonymous

- katieb

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- jim_thompson5910

Start off by defining variables

- jim_thompson5910

let x = amount of peanuts (in pounds), y = amount of almonds (in pounds) and z = amount of raisins (in pounds)

- anonymous

okey (:

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## More answers

- jim_thompson5910

If
"A food store makes a 11–pound mixture of peanuts, almonds, and raisins"
then what does this mean?

- anonymous

That they add up to 11 pounds haha

- jim_thompson5910

good, so x y and z add to 11, which means
x+y+z = 11

- jim_thompson5910

that's one equation

- anonymous

so x+y+z=11 :o

- jim_thompson5910

yes

- anonymous

But then we have to factor in how much they cost..

- jim_thompson5910

We also know that
"The cost of peanuts is $1.50 per pound, almonds cost $3.00 per pound, and raisins cost $1.50 per pound"
So x lbs of peanuts cost 1.5x dollars, y lbs of almonds cost 3y dollars, and z lbs of raisins cost 1.5z dollars
They total 1.5x + 3y + 1.5z dollars

- jim_thompson5910

this total cost is given to be $21
so
1.5x + 3y + 1.5z = 21

- anonymous

so like, 1.50x+3.00y+1.50=21..

- anonymous

Haha Okey you already said that x.x my bad

- anonymous

1.50z* I meant btw haha. I get that indeed.

- jim_thompson5910

close, you mean to say 1.50x+3.00y+1.50z=21
but you got it

- anonymous

And don't forget twice the peanuts!

- jim_thompson5910

that's your 2nd equation

- jim_thompson5910

twice the peanuts as what?

- anonymous

it says, the recipe calls for twice as many peanuts as almonds O.o

- jim_thompson5910

good, so the equation for that is?

- anonymous

ermm.. I dunno lol.. x(times2y)+y+z=11 O.O I have no idea.

- jim_thompson5910

not quite

- anonymous

knew it was wrong x.x

- jim_thompson5910

Amount of peanuts = 2 times (amount of almonds)

- jim_thompson5910

y = 2 times x
or just
y = 2x

- anonymous

oh pshh I should have thought of that x.x

- jim_thompson5910

so you have these 3 equations
x+y+z = 11
1.5x + 3y + 1.5z = 21
y = 2x

- jim_thompson5910

you can use them to solve for x, y and z

- jim_thompson5910

I'll get you started

- anonymous

Lol I can't because I'm stupid, But smart people definitely can haha

- jim_thompson5910

x+y+z = 11
x+2x+z = 11
3x+z = 11
---------------------
1.5x + 3y + 1.5z = 21
1.5x + 3(2x) + 1.5z = 21
1.5x + 6x + 1.5z = 21
7.5x + 1.5z = 21

- jim_thompson5910

so after plugging in y = 2x, you now have the new system
3x+z = 11
7.5x + 1.5z = 21

- anonymous

O.o That all made absolutely no sense to me but it seems to for you and that's good enough for me

- jim_thompson5910

where are you lost?

- jim_thompson5910

I took
x+y+z = 11
plugged in y = 2x (ie replaced every 'y' with '2x') to get
x+2x+z = 11
then I combined like terms to get
3x+z = 11

- jim_thompson5910

I did the same with 1.5x + 3y + 1.5z = 21 to get 7.5x + 1.5z = 21

- anonymous

oh okay ! I see now, sorry it's really late and I'm super tired :/

- jim_thompson5910

so you have
3x+z = 11
7.5x + 1.5z = 21
now

- jim_thompson5910

solve the first equation for z to get this
3x+z = 11
3x+z - 3x = 11 - 3x
z = 11 - 3x

- anonymous

Alrighty

- jim_thompson5910

Since z = 11 - 3x, you can plug it into the other equation and solve for x
7.5x + 1.5z = 21
7.5x + 1.5(11 - 3x) = 21
Solve for x

- jim_thompson5910

oh wait, i made a variable mistake...i switched x and y
one sec

- jim_thompson5910

Let me fix real quick

- anonymous

I was thinking that haha x) ..

- anonymous

uhh ? your comment disappeared .. :o

- jim_thompson5910

it should be x = 2y, not y = 2x
so
x+y+z = 11
2y+y+z = 11
3y + z = 11
z = 11 - 3y
---------------------------
1.5x + 3y + 1.5z = 21
1.5(2y) + 3y + 1.5z = 21
3y + 3y + 1.5z = 21
6y + 1.5z = 21
===========================
6y + 1.5z = 21
6y + 1.5(11 - 3y) = 21
Solve that last equation for y

- jim_thompson5910

sry

- jim_thompson5910

Solve 6y + 1.5(11 - 3y) = 21 for y

- anonymous

haha it's okay

- anonymous

Aw man I have to go to bed soon I'm dead tired x.x

- jim_thompson5910

What do you get when you solve that equation for y

- anonymous

ergh I dunno x.x I'm brain dead

- jim_thompson5910

Then give it a break, get some sleep and you'll tackle it in the morning

- anonymous

Oh no I keep losing connection >.<

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